Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance

Mario Bukal

doi:10.3934/dcds.2021001

Article Contents

2021, Volume 41, Issue 7: 3389-3414. Doi: 10.3934/dcds.2021001

This issue Previous Article A constructive approach to robust chaos using invariant manifolds and expanding cones Next Article A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on

$ C^{1} $

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Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance

Mario Bukal

University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia

Received: December 31, 2019

Revised: September 30, 2020

Early access: January 2021

Published: July 2021

This work has been supported by the Croatian Science Foundation under Grant agreement No. UIP-05-2017-7249 (MANDphy) and in part by the bilaterial project No. HR 04/2018 between OeAD and MZO

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Abstract

In this paper we construct a unique global in time weak nonnegative solution to the corrected Derrida-Lebowitz-Speer-Spohn equation, which statistically describes the interface fluctuations between two phases in a certain spin system. The construction of the weak solution is based on the dissipation of a Lyapunov functional which equals to the square of the Hellinger distance between the solution and the constant steady state. Furthermore, it is shown that the weak solution converges at an exponential rate to the constant steady state in the Hellinger distance and thus also in the $ L^1 $-norm. Numerical scheme which preserves the variational structure of the equation is devised and its convergence in terms of a discrete Hellinger distance is demonstrated.

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Mathematics Subject Classification: Primary: 35B45, 35K30, 65M06, 65M12; Secondary: 35Q99, 65M15.

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Figure 1. Numerical evolution of the corrected DLSS equation for unit mass initial datum $ u_0 $ at different time moments: $ t_1 = 5\cdot10^{-6} $, $ t_2 = 4\cdot10^{-5} $, $ t_3 = 2\cdot10^{-4} $, and $ t_4 = 1.5\cdot10^{-3} $

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Figure 2. Errors with respect to time and space discretization parameters. Dashed lines indicate theoretical convergence rates of the numerical scheme

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Table 1. Numerical convergence rates: $ \kappa_t $ in time (left) and $ \kappa_s $ in space (right) calculated according to (67)

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